Theorem int0 4921

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4453	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4913	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 234	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 267	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2757	1 ⊢ ∩ ∅ = V

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450  ∅c0 4278  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-dif 4047  df-nul 4279  df-int 4907

This theorem is used by:  unissint  4931  uniintsn  4944  rint0  4947  intex  5304  intnex  5305  oev2  8532  fiint  9319  elfi2  9407  fi0  9413  cardmin2  10081  00lsp  21257  cmpfi  23727  ptbasfi  23901  fbssint  24158  fclscmp  24350  zarcmplem  34556  rankeq1o  37007  bj-0int  38097  heibor1lem  38819  ipoglb0  50178  mreclat  50181